Geodesically Complete Metrics and Boundary Nonlocality in Holography: Consequences for the Entanglement Entropy
Abstract
We show explicitly that the full structure of IIB string theory is needed to remove the nonlocalities that arise in boundary conformal theories that border hyperbolic spaces on AdS. Specifically, using the Caffarelli/SilvestriCaffarelli and Silvestre (2007), Graham/ZworskiGraham (2001), and Chang/GonzalezChang and González (2011) extension theorems, we prove that the boundary operator conjugate to bulk pforms with negative mass in geodesically complete metrics is inherently a nonlocal operator, specifically the fractional conformal Laplacian. The nonlocality, which arises even in compact spaces, applies to any degree pform such as a gauge field. We show that the boundary theory contains fractional derivatives of the longitudinal components of the gauge field if the gauge field in the bulk along the holographic direction acquires a mass via the Higgs mechanism. The nonlocality is shown to vanish once the metric becomes incomplete, for example, either 1) asymptotically by adding N transversely stacked Ddbranes or 2) exactly by giving the boundary a brane structure and including a single transverse Ddbrane in the bulk. The original Maldacena conjecture within IIB string theory corresponds to the former. In either of these proposals, the location of the Ddbranes places an upper bound on the entanglement entropy because the minimal bulk surface in the AdS reduction is illdefined at a brane interface. Since the brane singularities can be circumvented in the full 10dimensional spacetime, we conjecture that the true entanglement entropy must be computed from the minimal surface in 10dimensions, which is of course not minimal in the AdS reduction.
I Introduction
Intrinsic to Maldacena’s conjecture Maldacena (1999) that supergravity (and string theory) on dimensional space () times a compact manifold, a sphere in the maximally supersymmetric case, is equivalent to the large limit of conformal field theory in dimensions is the separation between bulk and boundary physics. The impetus for such ideas originates from pioneering work of SusskindSusskind (1995) and ’t Hooft’t Hooft (1993), who named the bulkboundary correspondence in gravity holography. However, as is well knownWitten (1998), the boundary of any asymptotically AdS spacetime lives at infinity. Hence, it does not inherit a well defined metric structure. The structure it does acquire at the boundary is entirely conformal as can be seen from the Euclidean signature rendition
(I.1) 
of the AdS metric. The singularity at can be removed by considering the conformally equivalent metric . In fact, any metric of the form
(I.2) 
would do the trick ( a real function) , thereby laying plain the inherent conformal structure of the boundary.
Hence, correlation functions of the conformal operators of the boundary theory should in principle encode the physics of quantum gravity in a spacetime that is asymptotically AdS. Strictly speaking, however, the conformal field theory (CFT) only describes the physical excitations near the boundary. Precisely how far into the bulk this descriptionKabat et al. (2011) applies remains an open question. A key aspect of the mapping is that the CFT contains local operators. Consider the example of a free field propagating in the bulk that obeys the KleinGordon equation. The correspondence between the bulk and the boundary physics stems potentially from the equivalence between the partition functions
(I.3) 
in the two theories where is the boundary operator, is the extension of the bulk field to the boundary and is the supergravity partition function averaged over all doublepole metrics. This form of the correspondence relies on an integration of the bulk action by parts and then an evaluation of the corresponding boundary termsGubser et al. (1998); Witten (1998). Alternatively, an equivalence can be established by extrapolating the behavior of bulk correlators to the boundaryBanks et al. (1998). Near the boundary, the solutions scale asymptotically as
(I.4) 
where is the holographic coordinate (so that the conformal boundary is at ) and , where is a number such that , and . The AdS/CFT dualityBanks et al. (1998); Polchinski (2010) in the extrapolation scheme dictates that we associate with this boundary behavior a corresponding local conformal operator whose dimension is . So, setting , the precise prescriptionPolchinski (2010) for accomplishing this is the limit
(I.5) 
where the boundary nonrenormalizable term has been removed so that the limit is well defined. One should think of as the BDHMBanks et al. (1998) formulation of the AdS/CFT correspondence:
(I.6) 
In this work, we show that when the bulk action, , is a Gaussian theory, then for some values of the mass squared of the bulk field , the operator augmenting the boundary theory is an antilocal operator: the fractional Laplacian. This is true regardless of the formulation that is used to express the bulkboundary correspondence. We then argue – following a logic reminiscent of the one adopted by Giddings (cf. Giddings (1999) and Harlow and Stanford (2011))–that since interactions turn off near the boundary, even in an interacting theory, the operator must still be an antilocal operator (presumably having the fractional Laplacian as leading term).
Another way in which the interaction terms tend to vanish is for the theory with large. In the bulk interactive theory the bulk action can only be calculated by perturbative expansion, e.g. using Witten’s graphs. Nonetheless, as shown in Lee et al. (1998) and Arutyunov and Frolov (2000), when the interacting fields correspond to KaluzaKlein modes of the compactified supergravity theory, say , then, if we write the action as,
(I.7) 
then the coefficients . This indicates that, as , the operators in the boundary which are dual to KaluzaKlein modes must behave as the fractional Laplacian, for suitable values of the mass squared.
The mathematics behind either the original or the BDHMBanks et al. (1998) AdS/CFT correspondence is that of determining the asymptotic structure of solutions to the equations of motion, when approaching the conformal boundary. Three mathematical groupsCaffarelli and Silvestre (2007); Graham (2001); Chang and González (2011) have developed theorems that solve such boundary extension problems. We first show that Eq. (I.5) is explicitly of the form needed to apply the Caffarelli/Silvestre extension theorem. Applying the theorem allows us to show that for a bulk field obeying the KleinGordon equation, Eq. (I.5) is explicitly the fractional Laplacian acting on the boundary field . Unlike the normal Laplacian, the fractional Laplacian is explicitly a nonlocal operation in that it requires knowledge of the function everywhere for it to be evaluated. Within the AdS/CFT conjecture as a wholeMaldacena (1999); Gubser et al. (1998); Witten (1998); Banks et al. (1998), our work establishes a technical procedure for going between bulk fields defined by appropriate equations of motion and corresponding operators at the conformal boundary. Although using the Caffarelli/Silvestre theorem requires that we equate the bulk field, namely the scalar field in with in , the final result is more than a field redefinition. This procedure results in a closed expression for the operator at the conformal boundary. The difference with our result and the claim in the original conjecture is that is explicitly nonlocal. The key results are summarized in Table 1. We demonstrate that the nonlocality is an intrinsic property of spacetimes that are geodesically complete. We show explicitly that the Maldacena conjecture that local theories lie at the boundary of AdS spacetimes is recovered only if some degree of geodesic incompleteness is present in the bulk metric. For example, stacking N branes transverse to the radial direction in the bulk leads to a local theory at the boundary in the asymptotic limit of . As a result, our work implies that the strong form of the AdS/CFT duality with finite cannot hold without including nonlocal operators at the boundary. Implicit in any form of the conjecture is the fixing of a vacuum in the boundary and the bulk. While in general the two vacua might not be related, in our work the boundary vacuum emerges from the bulk. This type of emergence is physical and explicit. In order to connect our formulation of the AdS/CFT correspondence with the more standard strongweak formulation one must understand the interconnection between our vacuum and the one of N=4 SYM. This will be addressed in a later publication. Geodesic incompleteness poses a problem for the geometric interpretationRyu and Takayanagi (2006) of the entanglement entropy as the minimal surface cannot cross a Ddbrane singularity. We propose that a higherdimensional geometric construction within the full type IIB string theory is necessary to retain the minimal surface idea. Boundary nonlocalities appear as well for vector fields where the fractional exponent is governed by the mass of the bulk gauge field along the holographic direction. This provides an explicit mechanism for producing anomalous dimensionsLimtragool and Phillips (2016) for boundary gauge fields.
Bulk Operator  Boundary Operator: 

Ii Preliminaries: Evaluation of Eq. (i.5)
To put our work in the context of the AdS/CFT correspondence, we review the standard procedureWitten (1998) for a massive scalar field. To this end, we work with the action
(II.1) 
For the purposes of this initial discussion, we assume an AdS background with Euclidean signature (although we can and will work, mutatis mutandis, with the general case of a black hole endowed with near horizon AdS geometry). The equations of motion for the field are then simply given by,
(II.2) 
where is the Laplacian. It is a classical fact that this equation admits the existence of a unique solution on with any given boundary value (the boundary is a sphere , as described by copies of given by and the point ).
As is standardWitten (1998), we assume that in the correspondence between and conformal field theory on the boundary, should be considered to couple to a conformal field , via: . Thus, in order to compute the two point function of , one must evaluate for a classical solution with boundary value . The equation of motion can be rewritten in the form,
(II.3) 
where is such that (i.e., ). Thus, as shown in Mazzeo and MelroseMazzeo and Melrose (1987), such a solution has the form,
(II.4) 
unless belongs to the pure point spectrum of . Here and are functions on the conformal boundary . A vast generalization of this fact, which we shall use later, can be found in Mazzeo and Melrose (1987). We refer to as the restriction of to the boundary of . Operationally, the formal AdS/CFT correspondence can be established by taking the finite part of the result of integrating by parts,
(II.5) 
where denotes the finite part of the divergent integral and is the volume form of . Therefore, must be the twopoint function of .
We claim at this point that is indeed the Riesz fractional Laplacian
In order to show this, we need to appeal to a construction due to Caffarelli and Silvestre Caffarelli and Silvestre (2007), which characterizes the Riesz fractional Laplacian of a function defined on via an extension problem. Explicitly, what they showed is that given a function defined on , a solution to
(II.6)  
(II.7) 
has the property that
(II.8) 
for some (explicit) constant only depending on and .
Now we observe that if solves the massive problem (II.2) (in fact its representation in the form of Eq. (II.3)), then an easy computation shows that the function
(II.9) 
solves the CaffarelliSilvestri extension problem, Eqs. ((II.6)) and ((II.7)). But since a solution, , to the massive problem has the asymptotic expansion (using that ),
(II.10) 
it then follows that
(II.11) 
Now we make two observations. On the one hand by the asymptotic expansion of above, it must be that
(II.12) 
On the other, by the result of Caffarelli and SilvestriCaffarelli and Silvestre (2007) this limit is , up to a constant factor, thus showing that the two point function of the operator is a multiple of .
To address the BDHM formulationBanks et al. (1998), we note that the Cafferelli/Silvestre extension equation, Eq. (II.12), is precisely of the form of the operator identity in Eq. (I.5). To make this more transparent, we note that powers of can be traded for derivatives with respect to since Eq. (II.12) is based on the asymptotic expansion of the solutions for the equations of motion, namely Eq. (II.4). Consequently, we rewrite Eq. (I.5) as
(II.13) 
With the substitution , this equation is precisely of the form of Eq. (II.12) thereby offering another proof that the fractional Laplacian is the operator dual of the bulk free scalar field. We note that the fractional Laplacian in flat space is a conformal operator and in fact this is a general feature of operators obtained in this fashion via a scattering process (see Eq. IV.20). This should be kept in mind as one considers the true boundary of AdS, which is a sphere in Euclidean signature.
It is remarkable that one attains all nonnegative real values of , even in this nonconformal picture, as is allowed by the BreitenlohnerFreedman (BF)Breitenlohner and Freedman (1982) bound for stability. However, the theory thus presented has the unfortunate feature of being nonlocal however. As we will see such nonlocality is unavoidable and present even in the conformal construction. One should observe that the negative BF bound is of course only possible for states such that is not finite (otherwise the mass term would have to be positive). In fact, in complete generality, J. Lee (Lee (1995)) proved that the essential spectrum of any asymptotically Einstein manifold is bounded from below by and in the noncompact case, there are no embedded eigenvalues, thus ensuring that there are no squarenormalizable (i.e., renormalizable) eigenfunctions.
Iii Stability and Planck length
Of course, the stability condition depends on the AdS radius of curvature, . To introduce this length, we rescale the metric,
(III.1) 
accordingly. Using this Lorentzian metric, we write the action for the KleinGordon field as
(III.2) 
Therefore, the relevant equation to establish the asymptotic structure of Eq. (II.4), is
(III.3) 
Performing the change of variablesKleban et al. (2005) and setting , we obtain
(III.4) 
which shows that the Hamiltonian is a sum of squares (up to adding the boundary term ) provided that
(III.5) 
which is the BF bound. This of course still makes possibly arbitrarily close to , but as grows, this occurs with a mass terms which are increasingly close to ,
(III.6) 
Therefore, the non local phenomenon present at the boundary theory requires the presence of less strange (tachyonic) matter in the bulk as the black hole radius increases.
Iv Conformal Holography
Away from flat space, the fractional Laplacian is not a conformal operator. To ensure conformality, we must include a conformal sector in the starting action. Namely, we consider the following action in the bulk
(IV.1) 
where the standard EinsteinHilbert action (with the GibbonsHawking boundary term) is given by
(IV.2) 
, is the induced metric on and is the trace of the extrinsic curvature of the boundary. The new term is something we name conformal matter given by the action
(IV.3) 
with
(IV.4) 
The new term in , , contributes to the EulerLagrange equations in the form of the conformal Box operator,
(IV.5) 
The advantage of using a conformal action (as part of the total action) is that one incorporates the fact that the boundary only has a well defined conformal class of metrics (arising from conformally compactifying AdS) into the theory. The boundary theory operators naturally correspond to conformal Laplacians. Moreover, in the case of a conformal Einstein manifold (such as the hyperbolic space), simplifications arise. Recall that on a Riemannian manifold of dimension , the conformal Laplacian is
(IV.6) 
which, after a conformal change of metric, , transforms as
(IV.7) 
For the hyperbolic metric , the scalar curvature is , so that
(IV.8) 
and now, the BFBreitenlohner and Freedman (1982) stability bound becomes . This condition actually is independent of the dimensionality because we can write with which is equivalent to . The conformal dimension of the field is exactly .
In complete generality, one defines an asymptotically AdS spacetime as a dimensional space time such that has a topological boundary characterized as follows:

There exists a function in such that and (i.e. is a defining function for the boundary,

is a smooth Lorentzian metric,

(The space looks like AdS at infinity) there exists a diffeomorphism and real numbers (here is the coordinate/defining function of the boundary on AdS) such that for , and

satisfies the Einstein equations: .
In this context, we still propose an AdS/CFT type correspondence, but with the Lagrangian given by the conformal matter equation above, namely Eq. ((IV.4)). Now the correspondence requires that we find solutions to the classical equations of motion,
(IV.9) 
and then perform the same scattering process we useded earlier in the classical theory. In general, due to the presence of the potential part , this analysis is considerably more complicated than the one performed in the classical case for AdS, and it tends to be very different even from the classical case of asymptotic AdS gauge/gravity duality (where one merely studies classical solutions of motion: )
Nonetheless, this theory becomes considerably easier in the case that . In this case, again switching to Euclidean signature, we can infer from the Einstein equation that the scalar curvature has to be constant which we normalize such that,. In this circumstance, the classical equations of motion for conformal matter (i.e., Eq. (IV.9)) reduce to,
(IV.10) 
We write yet again this equation in the form,
(IV.11) 
where is such that (i.e., ). Thus, setting , such a solution has the form,
(IV.12) 
Here and are functions on the conformal boundary . Next, as in the case of the classical Laplacian (as opposed to the conformal one we are analyzing here), if we set
(IV.13) 
one readily finds that solves the CaffarelliSilvestri extension problem, Eqs. (II.6) and (II.7). It is now plain that,
(IV.14) 
and that by the asymptotic expansion of above,
(IV.15) 
In the general case, we consider the asymptotic solutions to Eq. (IV.9) and define the scattering operator as follows. Solutions to
(IV.16) 
have the form
(IV.17) 
for all unless belongs to the pure point spectrum of . The scattering operator on is defined as .
Following Chang and González (2011), we define the conformally covariant fractional powers of the Laplacian (on the conformal boundary) as
(IV.18) 
for , , . One readily sees that , where is a pseudodifferential operator of order .
By the property of proven in Graham (2001), one has,
(IV.19) 
where,
(IV.20) 
and is the conformal infinity and
(IV.21) 
In this context, the operator is found using,
(IV.22) 
Consequently, the corresponding boundary operator is which persists under any change to the bulk metric as long as the conformal boundary remains unchanged.
iv.1 A few words on the conformal Laplacian
A choice of a Lorenzian metric
(IV.23) 
that connects two vector bundles. Given a vector bundle endowed with a Hermitian connection , more generally one defines the conformal Laplacian as
(IV.24) 
where
(IV.25) 
and after suitably trivializing sections of we can think of it as an operator with
(IV.26) 
In this paper, we will suppress the line bundles, , by fixing a conformal representative.
V Gauge Theory
In order to follow WittenWitten (1998) closely, we again switch to Euclidean signature. Here we consider adding the Lagrangian,
(V.1) 
where is the filed strength of the 1form . The classical equations of motion are then (equivalent to) Maxwell’s,
(V.2) 
and in fact, the previous scattering process can be repeated, mutatis mutandis, as follows. Given a 1from on the conformal boundary, we want to solve for solutions to
(V.3)  
(V.4) 
It is a standard consequence of the Weitzenböck formula, which relates the Hodge Laplacian to the standard Laplacian, that the previous equation is related to
(V.5) 
v.1 Higgs Mechanism for Fractional Gauge Fields
In this section we describe how the process of symmetry breaking along the holographic direction gives rise to fractional Laplacians acting on Gauge fields at the boundary. We describe here for simplicity just the case in which the gauge group is . We consider the Lagrangian,
(V.6) 
where is a function only of the radial coordinate, . This Lagrangian is invariant not only under the transformation but also the complexified gauge group Witten (1991). For generality, we consider this larger gauge group here as it generates negative masses of the gauge field. Hence, we consider a transformation of the form,
(V.7) 
with , where can be complex. As is standard, we expand around the vacuum expectation,
(V.8) 
In other words, we break the symmetry in the radial direction by writing,
(V.9) 
where is merely a function of the holographic direction. Then the standard symmetry breaking,
(V.10) 
produces the Lagrangian
(V.11) 
We can now apply the previous analysis to obtain terms of the kind at the boundary, where .
Observe that Eq. (V.10) shows that the for (i.e. in the nonholographic directions). Therefore we have clearly broken the symmetry merely in the holographic direction, thus leaving the boundary theory free to have any type of symmetry we please. Consequently, we have provided a mechanism for understanding how boundary theories proposed recentlyDomokos and Gabadadze (2015); Limtragool and Phillips (2016) acquire gauge fields with fractional dimensions. Results for the boundary form of the operators is summarised in Table 1.
Vi Branes in Action: Maldacena’s Duality on Incomplete Metrics
How do we then recover Maldacena’s conjecture that local conformal theories lie at the boundary of AdS spacetimes? A crucial detail in the MaldacenaMaldacena (1999) construction based on type IIB string theory is the D3 branes which he stacked transversely in the bulk. We show explicitly here that it is only from these branes in the asymptotic limit that the gaugegravity correspondence is free of nonlocal interactions. That is, only when such branes are retained does the conformal theory field theory on the boundary have explicitly local operators.
Recall from Horowitz and StromingerHorowitz and Strominger (1991) that there is a black brane solution of IIB string theory which is spherically symmetric. Part of the low energy action from string theory is given explicitlyHorowitz and Strominger (1991) by
(VI.1) 
where is a closed form. We take and the extremal solution (with no event horizon) is given by
(VI.2) 
where here is the number of stacked branes (or flux of the black hole), is the string tension and the coupling constant. We now observe that the appears as a rescaling of the AdS metric and the rescaling property (Eq. (IV.7)), yields
(VI.3) 
whence we derive that the equations of motion in the metric are equivalent to
(VI.4) 
with the masssquared in the theory (bounded from below by the BF boundBreitenlohner and Freedman (1982)) thus showing that the boundary fractional Laplacians are of the type with . Since , this shows that strictly as , the nonlocalities disappear. This proves our assertion that a conformal theory with purely local operators obtains only in the limit of an infinite number of transversely stacked branes.
Alternatively, consider the string IIB solution whose background metric we write in general form as
(VI.5) 
where and . The metric is on for some Einstein 6manifold . The equations of motion dictate for to be a function of the transverse coordinates satisfying
(VI.6) 
where is the density of Ddbranes. For instance, the standard solution is obtained by choosing with as above, so that is a delta function counted with multiplicity determined by (hence the description of it as a stacking of branes positioned at the ”horizon” ). In this application we take to be a harmonic function that has a brane singularity at and another transverse brane somewhere in the bulk at (these are strictly speaking walls as they are codimension 1). We are interested in the limit in which the Dbrane approaches the boundary as illustrated in Fig. (1); that is, . It is clear from the description of the singularity of the Laplacian of that near the singularity, is an absolute value singularity. It is then easy to construct solutions of this type that exhibit a full symmetry in the limiting configuration.
We can make this supergravity argument come to light in a simple example of the RandallSandrumRandall and Sundrum (1999) type of metric in which the absolute value singularity is explicitly manifest. Our argument works perfectly well in the IIB supergravity model, but for the sake of expository clarity we present this simpler model instead. We consider the 5dimensional spacetime with , which we think of as a fluctuation of the directions of the RandallSundrum metric where is a length scale depending only on the mass (the analogue of the Planck mass) and the (negative) cosmological constant. The presence of in the exponential guarantees that the metric is geodesically incomplete. Such incompleteness has no affect on the connectedness of the boundary as guaranteed by the WittenYau theoremWitten and Yau (1999); Reid and Wang (2012). As in RandallSundrumRandall and Sundrum (1999) we consider the direction to take values in the quotient of the circle (which we think of as the interval with the points and identified). Then, since the coefficients of the metric at are , the effective action of a massive particle at the brane positioned at is proportional to
(VI.7) 
where . This clearly shows that for sufficiently large, the negative (effective) mass terms again become asymptotically positive, thereby leading to a vanishing of the scalar solutions which give rise to the nonlocality. The largeness of is of course an indication of a wall singularity which causes a ”warping” of the compact manifold, in the language of Chan et al. (2000). We see explicitly then that incompleteness coupled with a wall singularity are needed to rid the boundary theory of nonlocality.
This argument can be generalized beyond the RandallSundrum metric. In hyperbolic space, the mass of a string joining the two branes grows quadratically as . Once , the mass becomes positiveZwiebach (2009). It is this mass that sets the scale for the masses of bulk scalar fields. The solutions to the scalar field equations of motion we found earlier which give rise to the nonlocal boundary interactions are no longer valid should be sufficiently small so that the mass is positive, that is, a violation of the BF boundBreitenlohner and Freedman (1982). Hence, any type of Ddbrane placed transverse to the holographic direction in a geometry in which the boundary is viewed as a brane singularity is sufficient to kill the nonlocal interactions found here. The essence of this argument is that transverse walls break the completeness of the metric in the holographic direction. Once this completeness is broken, locality of the boundary theory obtains.
Vii Entanglement
Dbrane bulk singularities also affect the geometric interpretation of the entanglement entropyRyu and Takayanagi (2006). Computing the entanglement entropy of two regions in the boundary separated by a region simply requires delineating the bulk minimal surface on AdS that has as its Dirichlet boundary condition. Any such surface cannot remain minimal if it traverses a singularity in the bulk (see Fig. (1b)), such as a Dbrane. In the construction in Fig. (1), if the D3brane located at lies outside the minimal surface, the geometric interpretation of the entanglement entropy remains unaffected. However, as approaches the boundary, the minimal surface has to shrink to avoid the D3brane, thereby leading to a vanishing of the entanglement entropy in the limit . The singularity that arises in this limit depends on the type of D3branes that are in the 5dimensional theory. If the Dbrane arises from a reduction of a D3brane in the 10dimensional theory, then the brane stacking problem of MaldacenaMaldacena (1999) arises, which we treated previously. However, should the D3brane arise from a D9brane as in the previous section, then a wall singularity arises at the boundary resulting in a doubling of the metric. In this case, the metric resembles that of RandallSundrumRandall and Sundrum (1999) and, as a result, is incomplete. Interestingly, only in the noncompact limit, does the nonlocality vanish. Physically, this corresponds to completely separating the doubled regions of the metric off to opposing infinities. No entanglementvan Raamsdonk (2010) can arise in such a spacetime as the regions have each receded to infinities but in opposing directions.
Consequently, when the full brane structure of IIB string theory is considered, an alternative to the standard geometric interpretation of the entanglement entropy must be constructed. In the full 10dimensional structure, some singularities can be circumvented. Hence, we conjecture that the entanglement entropy should be constructed from the drawing the minimal mass (a type of current) in 10dimensional spacetime. The area of this surface we submit will be the true entanglement entropy. Note the projection of this surface to AdS does not preserve minimality because of the presence of curvature. We are advocating more than just an extension of the geometric interpretation of the entropy to AdS, where is a compact Einstein manifold, as has been done recentlyJones and Taylor (2016). What is required here is a generalization because singularities appear explicitly in the bulk.
Viii Closing Remarks
We have shown here that the full structure of IIB string theory is needed to remove the nonlocalities that arise in boundary conformal theories that border hyperbolic spaces. What this work ultimately tells us is that the gaugegravity duality as a statement about strictly hyperbolic spacetimes with complete metrics is not a theory about local conformal theories. The boundary theories contain fractional conformal Laplacians and hence are nonlocal. Consequently, the standard implementation of the gaugegravity duality, in which mechanisms such D3branes leading to metric incompleteness are absent, must yield local CFTs on the boundary. Metrics underlying the RandallSundrumRandall and Sundrum (1999) work are candidates for removing the nonlocalities.
Relatedly, all examples in which the gaugegravity correspondence has been worked out explicitly (and asymptotically explicit is included here), either D3branes (which we have shown remove the boundary nonlocality) are explicitly included in the bulkMaldacena (1999) or D3 branes are absent and the boundary theory contains explicitly nonlocal operatorsKitaev (); Sachdev (2015). On some level, this is not surprising because at the core of gravity are the equivalence principles which preclude local observables. As a result, any theory with gravity necessarily has less observables than a theory without it. Consequently, an a priori correspondence between a bulk theory of gravity and a local boundary CFT must include some added features in the bulk that would ultimately permit a purely local theory to emerge on the boundary.
Since there is no guarantee that the currentcarrying degrees of freedom in strongly correlated electron matter have a local description, the standard implementation of the gaugegravity correspondence without the inclusion of D3branes ultimately has utility. The nonlocal interactions that arise in this case can be useful in describing fractional gauge fields in strongly correlated quantum matter as in the strange metal of the cupratesLimtragool and Phillips (2016) or yield a method to obtain unparticle propagatorsGeorgi (2007); Domokos and Gabadadze (2015). In fact, the Higgs mechanism we have proposed here provides a general way of engineering boundary propagators with arbitrary anomalous dimensions. The precise form of the entanglement entropy in IIB string theory remains purely conjectural as of this writing.
Ix Appendix
Here we review some of the basics of the correspondence. For simplicity of notation, we consider the case . We fix with a given metric with fixed conformal infinity, which we take to be the conformal class of the round sphere . Of course, if we insist on being Einstein, this uniquely determines it as the classical AdS (this is still true if the conformal class is sufficiently close to the round oneGraham and Lee (1991)). Let be an effective action in the bulk. For instance this could be of the form,
(IX.1) 
We let be the Lagrangian of the boundary CFT. The primary operators at the boundary specify the spectrum of the said CFT. The correspondence dictates that one associates an operator at the boundary to a field in the bulk. The operator is associated with the source by
(IX.2) 
which determines the partition function to be
(IX.3) 
where is the given theory in the bulk evaluated on shell, so is an extension of satisfying the classical equations of motion. In order to calculate the (connected) npoint functions of the boundary theory, we write
(IX.4) 
and then calculate
(IX.5) 
We now specialize to the case where , the KleinGordon action. Since we are meant to calculate on shell, by integration by parts (eq. (II.5)), we find that the finite part of is
(IX.6) 
where we expand the classical solution as wehere as we demonstrate in the text. Therefore this determines and shows that there is no npoint function for . This computation holds also for any which is conformally compact, thus indicating that we have exactly determined the dual of the KleinGordon theory. To recover the full Maldacena duality one needs to add the Ddbrane constructions discussed in the text or perhaps other features of Type IIB string theory. As we demonstrate the nonlocalities vanish as .
Acknowledgements We thank G. Vanacore for carefully reading an earlier draft and the NSF DMR1461952 for partial funding of this project and the J. S. Guggenheim Foundation for providing a fellowship to P. W. P.
Footnotes
 The Riesz fractional Laplacian of a function defined on is where is some normalization constant.
 We could discuss here independently of the signature, but we choose, for definiteness of notation, to use a Lorenzian structure
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